| L(s) = 1 | + 3.23·3-s + 1.20·5-s + 7-s + 7.45·9-s − 2.69·11-s + 0.474·13-s + 3.90·15-s − 0.147·17-s − 1.75·19-s + 3.23·21-s + 5.23·23-s − 3.54·25-s + 14.4·27-s − 3.90·29-s + 0.792·31-s − 8.72·33-s + 1.20·35-s − 4.63·37-s + 1.53·39-s + 41-s + 2.14·43-s + 9.00·45-s − 5.64·47-s + 49-s − 0.476·51-s − 1.20·53-s − 3.25·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.540·5-s + 0.377·7-s + 2.48·9-s − 0.813·11-s + 0.131·13-s + 1.00·15-s − 0.0357·17-s − 0.403·19-s + 0.705·21-s + 1.09·23-s − 0.708·25-s + 2.77·27-s − 0.725·29-s + 0.142·31-s − 1.51·33-s + 0.204·35-s − 0.762·37-s + 0.245·39-s + 0.156·41-s + 0.327·43-s + 1.34·45-s − 0.822·47-s + 0.142·49-s − 0.0667·51-s − 0.165·53-s − 0.439·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.462553627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.462553627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 - 0.474T + 13T^{2} \) |
| 17 | \( 1 + 0.147T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 0.792T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 + 1.20T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 + 0.625T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 - 4.02T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.604404165604995950531119213518, −8.950380439767078468760542880867, −8.218576989920775259114432624267, −7.60004198263714752480049286225, −6.74371877112152573950088785448, −5.42697612510751214064544890991, −4.39963856893121206588914740091, −3.37486556929009035541107955958, −2.48634031350313504577618730417, −1.63247731740422318286133139218,
1.63247731740422318286133139218, 2.48634031350313504577618730417, 3.37486556929009035541107955958, 4.39963856893121206588914740091, 5.42697612510751214064544890991, 6.74371877112152573950088785448, 7.60004198263714752480049286225, 8.218576989920775259114432624267, 8.950380439767078468760542880867, 9.604404165604995950531119213518
